Mapping a numeric range onto another

the formula is

f(x) = (x - input_start) / (input_end - input_start) * (output_end - output_start) + output_start

I'll hook up this post here: https://betterexplained.com/articles/rethinking-arithmetic-a-visual-guide/ as it helped me a lot when trying to come up with this intuitively. Once you understand what the post is saying, it's trivial to come up with these formulas on your own. Note that I used to struggle with such questions as well. (I have no affiliations - just found it very useful)

say you have range [input_start..input_end], let's start by normalising it such that 0 is input_start, and 1 is input_end. this is simple technique to make the problem easier.

how do we do that? we'll, we'd have to shift everything left by input_start amount, such that if input x happens to be input_start, it should give zero.

so, let's say f(x) is the function that does the conversion.

f(x) = x - input_start

let's try it:

f(input_start) = input_start - input_start = 0

works for input_start.

at this point, it does not work for input_end yet, as we have not scaled it.

let's just scale it down by the length of the range, then we'll have the biggest value (input_end) mapped to one.

f(x) = (x - input_start) / (input_end - input_start)

ok, let's give it a try with input_end.

f(input_end) = (input_end - input_start) / (input_end - input_start) = 1

awesome, seems to work.

okay, next step, we'll actually scale it to output range. It is as trivial as just multiplying with the actual length of the output range, as such:

f(x) = (x - input_start) / (input_end - input_start) * (output_end - output_start)

now, actually, we're almost done, we just have to shift it to right so that 0 starts from output_start.

f(x) = (x - input_start) / (input_end - input_start) * (output_end - output_start) + output_start

let's give it a quick try.

f(input_start) = (input_start - input_start) / (input_end - input_start) * (output_end - output_start) + output_start

you see that the first part of equation is pretty much multiplied by zero, thus cancelling everything out, giving you

f(input_start) = output_start

let's try input_end as well.

f(input_end) = (input_end - input_start) / (input_end - input_start) * (output_end - output_start) + output_start

which in turn will end up as:

f(input_end) = output_end - output_start + output_start = output_end

as you can see, it now seems to be mapped correctly.

Let's forget the math and try to solve this intuitively.

First, if we want to map input numbers in the range [0, x] to output range [0, y], we just need to scale by an appropriate amount. 0 goes to 0, x goes to y, and a number t will go to (y/x)*t.

So, let's reduce your problem to the above simpler problem.

An input range of [input_start, input_end] has input_end - input_start + 1 numbers. So it's equivalent to a range of [0, r], where r = input_end - input_start.

Similarly, the output range is equivalent to [0, R], where R = output_end - output_start.

An input of input is equivalent to x = input - input_start. This, from the first paragraph will translate to y = (R/r)*x. Then, we can translate the y value back to the original output range by adding output_start: output = output_start + y.

This gives us:

output = output_start + ((output_end - output_start) / (input_end - input_start)) * (input - input_start)

Or, another way:

/* Note, "slope" below is a constant for given numbers, so if you are calculating
   a lot of output values, it makes sense to calculate it once.  It also makes
   understanding the code easier */
slope = (output_end - output_start) / (input_end - input_start)
output = output_start + slope * (input - input_start)

Now, this being C, and division in C truncates, you should try to get a more accurate answer by calculating things in floating-point:

double slope = 1.0 * (output_end - output_start) / (input_end - input_start)
output = output_start + slope * (input - input_start)

If wanted to be even more correct, you would do a rounding instead of truncation in the final step. You can do this by writing a simple round function:

#include <math.h>
double round(double d)
{
    return floor(d + 0.5);
}

Then:

output = output_start + round(slope * (input - input_start))

Arduino has this built-in as map.

Example:

/* Map an analog value to 8 bits (0 to 255) */
void setup() {}

void loop()
{
  int val = analogRead(0);
  val = map(val, 0, 1023, 0, 255);
  analogWrite(9, val);
}

It also has the implementation on that page:

long map(long x, long in_min, long in_max, long out_min, long out_max)
{
  return (x - in_min) * (out_max - out_min) / (in_max - in_min) + out_min;
}